Chern-Simons forms: interpretation and generalizations

Studying again differential geometry, anomalies and topology, I wondered if there is ANY physical interpretations (in terms of QFT or even classical field theory) of the Chern-Simons forms, via vacuum, physical processes or alike. That is, if

$$\omega_{2n+1}=n\int_0^1 dt t^{n-1}Tr[A(dA+tA^2)^{n-1}]$$

Also related, the shift of the 1-form $$A\rightarrow A+v$$ seems to have a relation with $$2n-2$$ forms and Faddeev-Popov ghosts fields, but as far as I know, CS forms can only exists in odd dimensions and, together YM, are the only gauge invariant actions. Is that true?